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Modified Bessel function of the third kind, with zero order.

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Interface

#include <codecogs/maths/special/bessel/k/k0.h>

using namespace Maths::Special::Bessel::K;

double K0(double x)
double K0_exp(double x )

Overview

Modified Bessel function, third kind, order zero.

The modified Bessel function of the third kind (or Weber Function) is a linear combination of Bessel functions of the first and second kinds. These are called the Hankel functions of the first and second kinds, and are one of two types of functions known as Hankel functions. The other type of Hankel function is defined by a contour integral.

The Bessel K , $K_v$ , is defined by the following second order differential equation:

$x \frac{\partial y (x)}{\partial x} + x^{2} \frac{\partial^{2} y (x)}{\partial x^{2}} -(x^{2})y =0$

(1)

FUNCTION

K0

Returns modified Bessel function of the third kind of order zero of the argument.

The range is partitioned into the two intervals [0,8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval.

Accuracy:

Tested at 2000 random points between 0 and 8. Peak absolute error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15. <pre> Relative error: domain # trials peak rms 0, 30 30000 1.2e-15 1.6e-16 </pre>

Example:

#include <stdio.h>
#include <codecogs/maths/special/bessel/k/k0.h>

int main()
{
for(double x=1; x<7; x+=1)
{
    double y=Maths::Special::Bessel::K::K0(x);
    printf("\n K0(%.1lf)=%lf", x,y);
}
return 0;
}

Output:

K0(1.0)=0.421024
K0(2.0)=0.113894
K0(3.0)=0.034740
K0(4.0)=0.011160
K0(5.0)=0.003691
K0(6.0)=0.001244

References

Cephes Math Library Release 2.8: June, 2000

Parameters

input argument

Author

Documentation by Nick Owens

Interactive Calculator

Result –